Radiative Squeezing Flow of Second Grade Fluid with Convective Boundary Conditions

Influence of magnetohydrodynamic (MHD) flow between two parallel disks is considered. Heat transfer analysis is disclosed due to thermal radiation and convective boundary condition. Appropriate transformations are invoked to obtain the ordinary differential system. This system is solved using homotopic approach. Convergence of the obtained solution is discussed. Variations of embedded parameters into the governing problems are graphically discussed. Skin friction coefficient and Nusselt number are numerically computed and analyzed. It is noticed that temperature profile is increasing function of radiation parameter.


Introduction
Squeezing flow has attracted significant attention of many researchers and scientists due to its paramount applications in various fields such as in bio-mechanics, food processing, mechanical, industrial and chemical engineering. This phenomenon is also observed in bearings, gears, rolling elements, machine tools, automotive engines, polymer processing, design of lubrication systems (including oil and grease systems) and injection and compression shaping etc. These type of flows are generated in many hydrodynamical machines and tools where vertical velocities or normal stresses are applying due to moving boundary. Stefan [1] was the first who studied the squeezing flows. He presented an adhoc asymptotic solution for flow of Newtonian fluid. Further Mahmood et al. [2] investigated the squeezed flow and heat transfer for viscous fluid towards a porous surface. Mustafa et al. [3] developed analytical solutions for the squeezing flow of nanofluid between the parallel disks. Qayyum et al. [4] examined the unsteady squeezed flow of Jeffery fluid between two parallel disks. Magnetohydrodynamic squeezing flow of second grade fluid between two parallel disks is studied by Hayat et al. [5]. Ganji et al. [6] investigated the magnetohydrodynamic (MHD) squeezed flow between two porous disks. Thermal radiation effects in time-dependent axisymmetric squeezing flow of Jeffery fluid between two parallel disks is analyzed by Hayat et al. [7]. Homotopy perturbation solution of MHD squeezing flow between two porous disks is studied by Domairry and Aziz [8].
Flow of electrically conducting fluid in presence of magnetic field is related to magnetohydrodynamics (MHD). The MHD systems are used in many applications such as power generators, accelerators, droplet filters, the design of heat exchangers, electrostatic filters, the cooling of reactors, pumps etc. Magnetic nanofluid is a fluid with unique characteristics of both liquid and magnetic field. Numerous applications involving magnetic nanofluids include drug delivery, contrast enhancement in magnetic resonance imaging and magnetic cell separation. Few representative studies on magnetohydrodynamics (MHD) nanofluid can be consulted through the investigations [9][10][11][12][13][14][15][16][17][18]. On the other hand the convective heat transfer has also mobilized substantial interest due to its significance in the industrial and environmental technologies including energy storage, gas turbines, nuclear plants, rocket propulsion, geothermal reservoirs and photovoltaic panels. The convective boundary condition has also attracted some interest and this usually is simulated via a Biot number in the wall thermal boundary condition. Recently Rashidi et al. [19] applied the one parameter continuous group method to investigate similarity solutions of magnetohydrodynamic (MHD) heat and mass transfer flow of viscous fluid over a flat surface with convective boundary condition. Analysis of heat and mass transfer in mixed convective flow towards a vertical flat surface with hydrodynamic slip and thermal convective boundary condition is examined by Rashidi et al. [20]. Bachok et al. [21] considered the stagnation point flow towards a shrinking or stretching surface with the bottom of surface heated through convection from a hot fluid. Magnetohydrodynamic (MHD) three-dimensional flow of an incompressible fluid induced by an exponentially stretching surface with convective boundary condition is investigated by Hayat et al. [22].
The radiation effects in the boundary layer flow is very important due to its application in physics, engineering and industrial fields such as glass production, furnace design, polymer processing, gas cooled nuclear reactors and also in space technology like aerodynamics rockets, missiles, propulsion system, power plants for inter planetary flights and space crafts operating at high temperatures. Hence thermal radiation effects cannot be ignored in such processes. Rosseland approximation is used to describe the radiative heat flux in the energy equation. Hayat et al. [23] examined the two-dimensional magnetohydrodynamic flow of thixotropic fluid towards a stretched surface with variable thermal conductivity and thermal radiation effects. Pal [24] examined the effects of Hall current in magnetohydrodynamic (MHD) flow and heat transfer characteristics of viscous fluid past an unsteady permeable radiative stretching surface. Bhattacharyya et al. [25] analyzed the rate of heat transfer in micropolar fluid flow towards a porous shrinking surface with thermal radiation. Heat transfer analysis in MHD flow of viscous fluid past an exponentially stretching sheet with suction/injection effects is conducted by Mukhopadhyay [26]. Hayat et al. [27] observed the mixed convection stagnation point flow of Maxwell fluid over a surface with convective boundary conditions and thermal radiation. Bhattacharyya [28] considered the MHD radiative flow of Casson fluid over a stretching sheet in the stagnation region. Sheikholeslami et al. [29] reported the effect of thermal radiation and heat transfer by considering two phase model in magnetohydrodynamic nanofluid flow.
The purpose of present paper is to analyze the analytic solution of squeezing flow of second grade fluid between two porous disks. Fluid is electrically conducting in presence of variable magnetic field. Heat transfer is carried out with thermal radiation and convective boundary condition. Series solutions are found by homotopy analysis method [30][31][32][33][34][35][36][37][38][39][40]. Velocity, temperature, skin friction coefficient and Nusselt number are analyzed for different emerging parameters. To check the validity of the solutions, we have presented the comparison of our results for limiting previously published papers [5,8]. An excellent agreement is found. with velocity dz dt ¼ À1 2 aHð1 À atÞ À1 2 while lower porous disk at z = 0 is stationary. Mathematically w ¼ @h @t represents the squeezing of the upper disk towards the lower disk. Variable magnetic field of inclined character via an angle ψ is applied. The law of conservation of mass, momentum and energy for the considered problem are @u @r þ @w @z þ u r ¼ 0; ð1Þ r @u @t þ u @u @r þ w @u @z @u @z @u @r þ 1 r @w @z @w @r þ @w @r subject to the boundary conditions In the above expressions u and w denote the velocity components in the r and z directions respectively, ρ the fluid density, σ the electrical conductivity, c p the specific heat, μ the dynamic viscosity, g Ã 0 the heat transfer coefficient at the lower plate, g Ã 1 the rate of heat transfer coefficient of fluid at upper plate, k the thermal conductivity far away from the disk, σ Ã the Stefan-Boltzmann constant and k Ã the mean absorption coefficient, w 0 corresponds to the case for suction while −w 0 leads to the case for blowing. It should be noted that the governing equations are reduced to viscous fluid when α = 0. We consider where T f is the temperature of the hot fluid at the lower permeable disk, T 1 the ambient temperature above the upper disk, T the fluid temperature such that T f > T and T 1 < T. The Eq (1) is identically satisfied and Eqs (2-6) take the forms where M denotes the magnetic parameter, S the squeezing parameter, α 1 the dimensionless second grade fluid parameter, R represents radiation parameter and A represents suction/blowing parameter.
The skin friction coefficient and the local Nusselt number are where @u @z þ @w @z @w @r þ @u @z @u @r À @w @r @u @r ; and z¼0;hðtÞ In dimensionless form we obtain in which Re r ¼ arHð1ÀatÞ is the local Reynold number, Nu r0 denotes the heat transfer rate at lower disk while Nu r1 is the heat transfer rate at upper disk. Similarly C fr1 and C fr0 represent the skin friction coefficient at upper and lower disks respectively.

Homotopic Solutions
In order to find the homotopy analysis solutions we choose the base functions: and express where a n and b n are the coefficients to be determined. The initial guesses (f 0 ,θ 0 ) and auxiliary linear operators (L f ,L θ ) are with C i (i = 1-6) denote the arbitrary constants.

mth-order deformation problems
The mth-order deformation problems are @ŷ m ðZ; pÞ @Z Z¼0 ¼ 0; @ŷ m ðZ; pÞ @Z ( For p = 0 and p = 1 one haŝ yðZ; 0Þ ¼ y 0 ðZÞ;ŷðZ; 1Þ ¼ yðZÞ: When p varies from 0 to 1 thenf ðZ; pÞ andŷðZ; pÞ start from the initial solutions f 0 (η) and θ 0 (η) and reach to the final solutions f(η) and θ(η) respectively. The values of auxiliary parameters is selected in such a manner that the series solutions converge. The general solutions (f m , θ m ) via special solutions ðf Ã m ; y Ã m Þ are where the C i (i = 1-6) are the involved constants.  Table 1 depicts that the series solutions are convergent up to six decimal places at 8 th order of approximation for momentum and 6 th order of approximation for temperature.

Convergence of solutions
Residual errors are calculated for momentum and energy equations by using expressions Figs 4 and 5 display the ℏ− curves for residual error of ƒ and θ in order to get the admissible range for ℏ. It is observed that correct result up to 6th decimal places is obtained by choosing the value of ℏ from this range. Further the series solutions converge in the whole region of η(0 < η < 1) when ℏ f = −0.9 = ℏ θ .

Discussion
In this subsection, we studied the influence of diverse parameters on the velocity, temperature, skin friction coefficient and local Nusselt number.

Velocity profile
Effect of magnetic parameter (M) on the velocity distribution f 0 (η) in cases of suction and blowing have been displayed through Figs 6 and 7. It is observed that the magnitude of the radial velocity f 0 (η) shows dual behavior with the increase in the magnetic parameter M. For higher . It is analyzed that the velocity distribution f 0 (η) near the porous walls decreases and suction effects are dominant (see Fig 8). The flow enhances since upper wall is moving towards the stationary porous wall and pressure is     In fact α 1 is inversely proportional to the viscosity. For larger fluid parameter α 1 the viscosity of fluid decreases and consequently the velocity distribution enhances. The behavior of angle of inclination ψ on the velocity profile f 0 (η) is illustrated in Fig 11. Here, f 0 (η) increases gradually when ψ enhances but there is a decrease in f 0 (η) when 0.4 η 1. It is due to the fact that by increasing angle of inclination ψ the influence of magnetic effects on fluid particle rises which enhances the Lorentz force. Therefore velocity distribution decreases. Also one can observed that ψ = 0 is the case when magnetic effect has no influence on the velocity distribution. For ψ = π/2 the fluid particles offered the maximum resistance.

Temperature distribution
Variation of magnetic parameter (M) on the temperature distribution θ(η) is shown in Fig 12. It is noted that temperature distribution increases for higher magnetic parameter M. This is due to the fact that an increase in M give rise to the resistive force (Lorentz force), and consequently the temperature distribution increases. Fig 13 illustrates the effect of squeezing parameter S on the temperature θ(η). This figure indicates that for larger values of squeezing parameter S the temperature distribution decreases. Higher values of squeezing parameter S indicate that kinematic viscosity decreases and depends upon the velocity and distance between disks. The effect of radiation parameter R on the temperature distribution θ(η) is visualized in Fig 14 by keeping other parameters fixed. Temperature distribution is an increasing function of radiation parameter R. The mean absorption coefficient decreases for higher thermal radiation parameter R. Therefore temperature distribution increases. Fig 15 is sketched to examine the influence of Prandtl number Pr on temperature profile. It is contemplated that an increase in Pr reduce the thermal boundary layer due to which the heat transfer rate enhances and as a result the temperature of fluid decreases. Fig 16 demonstrates the influence of angle of inclination (ψ) on temperature distribution θ(η). It is revealed that temperature profile θ(η) increases for higher values of inclination angle ψ. It is due to the fact that higher values of angle of inclination ψ corresponds to larger magnetic field which opposes the fluid motion. As a result the temperature profile θ(η) increases. Figs 17 and 18 display the impact of Biot number (γ 1 ,γ 2 ) on the temperature distribution θ(η). Fig 17 indicates the influence of Biot number (γ 1 ) at the lower disk on temperature distribution. It is observed that larger γ 1 leads to an increase in the temperature profile θ(η) and thermal boundary layer thickness. The Biot number is the ratio of the internal resistance of a solid to the thermal resistance of the disk surface. With the increase of the Biot number, the thermal resistance of the surface of disk decreases. Higher surface temperature is achieved due to increase in convection which makes the thermal effect to go deep in

Conclusions
Impact of thermal radiation in the squeezing flow of second grade fluid with convective boundary condition is explored. The following points of presented analysis are worthmentioning. • Velocity profile decreases while the temperature distribution increases for larger magnetic parameter.
• Effects of angle of inclination, Prandtl number and magnetic parameter are qualitatively similar.
• Effect of Prandtl number on temperature field is opposite to that of thermal radiation parameter R.
• Temperature distribution enhances for larger radiation parameter R.
• Effects of magnetic and squeezing parameters on velocity distribution having similar behavior for both suction and blowing cases.
• Skin friction coefficient increases for higher values of S, M, α 1 , and ψ.
• For the larger values of R, γ 1 , and γ 2 the magnitude of local Nusselt number increases at both upper and lower disks.